线性系统普遍速率下的近似解

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-09-25 DOI:10.1137/22m1517196

Stefan Steinerberger

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引用次数: 0

摘要

让它可逆，未知，已知。我们感兴趣的是近似解:这样的向量很小。我们证明了，对于所有的向量，存在由，其中的行生成的超平面上的正交投影的复合，它将原点映射到一个满足的向量。我们注意到这个上界与矩阵无关。这个过程是稳定的，因为。存在性证明是基于随机化Kaczmarz方法的概率细化分析，当求解高似然时，该方法似乎达到了这个速率。我们也证明了具有和满秩矩阵的一般版本。

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Approximate Solutions of Linear Systems at a Universal Rate

Let be invertible, unknown, and given. We are interested in approximate solutions: vectors such that is small. We prove that for all , there is a composition of orthogonal projections onto the hyperplanes generated by the rows of , where , which maps the origin to a vector satisfying . We note that this upper bound on is independent of the matrix . This procedure is stable in the sense that . The existence proof is based on a probabilistically refined analysis of the randomized Kaczmarz method, which seems to achieve this rate when solving for with high likelihood. We also prove a general version for matrices with and full rank.

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.